TITLE: Under what conditions does a ring R have the property that every zero divisor is a nilpotent element?
QUESTION [17 upvotes]: Under what conditions does a ring $R$ have the property that every zero divisor is a nilpotent element ? 
If we have a ring $R$, we know that every nilpotent element is either zero or a zero divisor.
If $A$ is the set of all nilpotents except the zero, and $B$ is the set of all nonzero zero divisors, then under what conditions on $R$ do we have $A = B$ ?

REPLY [3 votes]: If $A$ is a commutative ring with this property, then the zero-divisors form an ideal  of $A$ (since the nilradical is an ideal), in fact a prime ideal, and thus it's the unique minimal prime ideal of $A$.  Conversely, if $A$ has a unique minimal prime ideal containing all the zero-divisors, then this has to be the nilradical, so every zero-divisor is nilpotent.
People seem to be calling these rings 'primary rings,' since this condition characterizes the rings you get after quotienting an arbitrary ring by a primary ideal.  I don't know how widely used this term is.